Abstract
In this paper, a new Boundary Integral Equation (BIE) is proposed for solution of the scalar Helmholtz equation. Applications include acoustic scattering problems, as occur in room acoustics and outdoor and underwater sound propagation. It draws together ideas from the study of time-harmonic and transient BIEs and spatial audio sensing and rendering, to produce an energy-inspired Galerkin BEM that is intended for use with oscillatory basis functions. Pivotal is the idea that waves at a boundary may be decomposed into incoming and outgoing components. When written in its admittance form, it can be thought of setting the Burton–Miller coupling parameter differently for each basis function based on its oscillation; this is a discrete form of the Dirichlet-to-Neumann map. It is also naturally expressed in a reflectance form, which can be solved by matrix inversion or by marching on in reflection order. Consideration of this leads to an orthogonality relation between the incoming and outgoing waves, which makes the scheme immune to interior cavity eigenmodes. Moreover, the scheme is seen to have two remarkable properties when solution is performed over an entire obstacle: (i) it has a condition number of 1 for all positive-real wavenumber k on any closed geometry when a specific choice of cylindrical basis functions are used; (ii) when modelling two domains separated by a barrier domain, the two problems are numerical uncoupled when plane wave basis functions are used — this is the case in reality but is not achieved by any other BIE representation that the authors are aware of. Normalisation and envelope functions, as would be required to build a Partition-of-Unity or Hybrid-Numerical-Asymptotic scheme, are introduced and the above properties are seen to become approximate. The modified scheme is applied successfully to a cylinder test case: accuracy of the solution is maintained and the BIE is still immune to interior cavity eigenmodes, gives similar conditioning to the Burton–Miller method and iterative solution is stable. It is seen that for this test case the majority of values in the interaction matrices are extremely small and may be set to zero without affecting conditioning or accuracy, thus the linear system become sparse - a property uncommon in BEM formulations.
Highlights
Boundary Integral Equations (BIEs) can be an effective technique for solving wave scattering problems in homogeneous media
Despite the advantages of Boundary Element Method (BEM) – more straightforward meshing, a reduction in the number of degrees of freedom relative to methods that discretise the quantities in the volume and ease of modelling unbounded problems – its use in acoustic engineering has mostly been in a small set of specific cases
The configuration in [50] is interesting since the ‘testing’ integral is physically separated from the boundary on which the source densities are defined, making it immediately clear that non-uniqueness is associated with the ‘testing’ integral rather than the Kirchhoff–Helmholtz boundary integral equation (KHBIE)
Summary
Boundary Integral Equations (BIEs) can be an effective technique for solving wave scattering problems in homogeneous media. Despite the advantages of BEM – more straightforward meshing, a reduction in the number of degrees of freedom relative to methods that discretise the quantities in the volume and ease of modelling unbounded problems – its use in acoustic engineering has mostly been in a small set of specific cases These include, for example: prediction and optimisation of the sound scattering properties of acoustic objects and treatments [1,2] and noise barriers [3,4]; estimation of head-related transfer functions from head geometry [5,6]. Matrix compression techniques such as fast-multipole [9] and adaptive-cross-approximation [10] are capable of significantly reducing these requirements, but the scaling of computation cost and storage with frequency is still sufficiently unfavourable so as to preclude full audiblebandwidth simulation for most realistic sized room acoustics problems of interest
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